find the arc length of the particle circle

Answer:

3/8

Step-by-step explanation:

0.375 im hoping thats correct good luck

Answer:

The volume is increasing at a rate of 33929.3 cubic millimeters per second.

Step-by-step explanation:

Volume of a sphere:

The volume of a sphere of radius r is given by:

\(V = \frac{4\pi r^3}{3}\)

In this question:

We have to derivate V and r implicitly in function of time, so:

\(\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}\)

The radius of a sphere is increasing at a rate of 3 mm/s.

This means that \(\frac{dr}{dt} = 3\)

How fast is the volume increasing when the diameter is 60 mm?

Radius is half the diameter, so \(r = 30\). We have to find \(\frac{dV}{dt}\). So

\(\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}\)

\(\frac{dV}{dt} = 4\pi (30)^2(3) = 33929.3\)

The volume is increasing at a rate of 33929.3 cubic millimeters per second.

Answer:

1) \(\sqrt{53}(\cos286^\circ+i\sin286^\circ)\)

2) \(\displaystyle 5,-\frac{5}{2}+\frac{5\sqrt{3}}{2}i,-\frac{5}{2}-\frac{5\sqrt{3}}{2}i\)

Step-by-step explanation:

Problem 1

\(z=2-7i\\\\r=\sqrt{a^2+b^2}=\sqrt{2^2+(-7)^2}=\sqrt{4+49}=\sqrt{53}\\\\\theta=\tan^{-1}(\frac{y}{x})=\tan^{-1}(\frac{-7}{2})\approx-74^\circ=360^\circ-74^\circ=286^\circ\\\\z=r\,(\cos\theta+i\sin\theta)=\sqrt{53}(\cos286^\circ+i\sin 286^\circ)\)

Problem 2

\(\displaystyle z^\frac{1}{n}=r^\frac{1}{n}\biggr[\text{cis}\biggr(\frac{\theta+2k\pi}{n}\biggr)\biggr]\,\,\,\,\,\,\,k=0,1,2,3,\,...\,,n-1\\\\z^\frac{1}{3}=125^\frac{1}{3}\biggr[\text{cis}\biggr(\frac{0+2(2)\pi}{3}\biggr)\biggr]=5\,\text{cis}\biggr(\frac{4\pi}{3}\biggr)=5\biggr(-\frac{1}{2}-\frac{\sqrt{3}}{2}i\biggr)=-\frac{5}{2}-\frac{5\sqrt{3}}{2}i\)

\(\displaystyle z^\frac{1}{3}=125^\frac{1}{3}\biggr[\text{cis}\biggr(\frac{0+2(1)\pi}{3}\biggr)\biggr]=5\,\text{cis}\biggr(\frac{2\pi}{3}\biggr)=5\biggr(-\frac{1}{2}+\frac{\sqrt{3}}{2}i\biggr)=-\frac{5}{2}+\frac{5\sqrt{3}}{2}i\)

\(\displaystyle z^\frac{1}{3}=125^\frac{1}{3}\biggr[\text{cis}\biggr(\frac{0+2(0)\pi}{3}\biggr)\biggr]=5\,\text{cis}(0)=5(1+0i)=5\)

Note that \(\text{cis}\,\theta=\cos\theta+i\sin\theta\) and \(125=125(\cos0^\circ+i\sin0^\circ)\)

Answer:

I think the answer is she

The four rectangles can be painted with 7 different paints in 840 ways.

A permutation is an arrangement of objects in a definite order. The members or elements of sets are arranged here in a sequence or linear order. For example, the permutation of set A={1,6} is 2, such as {1,6}, {6,1}, there are no other ways to arrange the elements of set A.

The formula for permutattion is;

\(P_r_n = \frac{n!}{(n-r)!} \\\)

where r = 4

n = 7

Substituting the values for n and r

\(P = \frac{7!}{(7-4)!}\)

\(P = \frac{7!}{3!}\)

P = 840 ways.

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Answer:

-------------------------

Given two legs of a right triangle and we need to find the hypotenuse.

Use Pythagorean theorem:

The matching choice is C.

(a) The errors made by the student are:

Incorrectly expanding 49(2) as 24 instead of 98.

Mistakenly factoring the quadratic equation as (y - 8)(y - 1) instead of

\(y^{2} - 9y + 8.\)

(b) The exact solution to the equation is x = 7/11.

(a) The student made two errors in their solution:

Error 1: In the step \("22x + 49(2) = 0,"\) the student incorrectly expanded 49(2) as 24 instead of 98. The correct expansion should be 49(2) = 98.

Error 2: In the step \("y^{2} - 9y + 8 = 0,"\) the student mistakenly factored the quadratic equation as (y - 8)(y - 1) = 0. The correct factorization should be \((y - 8)(y - 1) = y^{2} - 9y + 8.\)

(b) To find the exact solution to the equation, let's correct the errors made by the student and solve the equation:

Starting with the original equation: \(22x + 4 - 9(2) = 0\)

Simplifying: 22x + 4 - 18 = 0

Combining like terms: 22x - 14 = 0

Adding 14 to both sides: 22x = 14

Dividing both sides by 22: x = 14/22

Simplifying the fraction: x = 7/11

Therefore, the exact solution to the equation is x = 7/11.

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Answer:

(b)

Step-by-step explanation:

You can clearly see that putting each factor equal to 0 , we get -6 , -2 , 3

Answer:

Step-by-step explanation:

Answer:

You can drive at the most 150 miles per day

Step-by-step explanation:

The value of m + n is given as follows:

A. 18.

The exponential expression for this problem is defined as follows:

\((x^5y^6)^{\frac{1}{5}}(x^3y^4}^{\frac{1}{4}}\)

Applying the power of power rule, we have that the exponents are given as follows:

Then the values of m and n are given as follows:

m = 7, n = 11.

Thus the sum is given as follows:

m + n = 7 + 11 = 18.

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The value of x in the equation is 16.

An equation is a mathematical expression that is made up of two expressions connected by an equal sign.

Therefore, let's solve the equation for variable x.

A variable is a number represented by letter in an equation.

Hence,

9 - (5x + 7) = - 2.8 - 4.7x

Therefore, let's open the bracket on the left side of the equation

9  - 5x - 7 = - 2.8 - 4.7x

2 - 5x =  - 2.8 - 4.7x

2 + 2.8x = -4.7x + 5x

4.8x = 0.3x

divide both sides of the equation by 0.3

x = 4.8 / 0.3

x = 16

Therefore,

x = 16

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Answer:Smith left 54 in the garden

Step-by-step explanation: 10% of 60 is 6

90% of 60 is 54

Hope this helps

The turning points of the function y = x³ - 3x² - 9x + 4 are (3, -23) and (-1, 9).

To determine the turning points of the given function y = x³ - 3x² - 9x + 4, we need to find the critical points where the derivative of the function is equal to zero.

1. Find the derivative of the function:

  y' = 3x² - 6x - 9

2. Set the derivative equal to zero and solve for x:

  3x² - 6x - 9 = 0

3. Factorize the quadratic equation:

  3(x² - 2x - 3) = 0

4. Solve the quadratic equation by factoring or using the quadratic formula:

  (x - 3)(x + 1) = 0

  This gives us two possible values for x: x = 3 and x = -1.

5. Substitute these critical points back into the original function to find the corresponding y-values:

  For x = 3:

  y = (3)³ - 3(3)² - 9(3) + 4

    = 27 - 27 - 27 + 4

    = -23

  For x = -1:

  y = (-1)³ - 3(-1)² - 9(-1) + 4

    = -1 - 3 + 9 + 4

    = 9

6. Therefore, the turning points are (3, -23) and (-1, 9).

Note: It appears that there was a typo in the original equation, where the term "-9x" should have been "-3x²". The above solution assumes the corrected equation.

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Answer:

Number 1 does not represent a function because it does not pass the Vertical Line Test.

Number 2 does not represent a function because it does not pass the Vertical Line Test.

Measure of angle E is, 25 degree

And, Measure of side FG is,

FG = 10 units

We have to given that;

In parallelogram EFGH,

EF = 3y

EH = 2 (y + 1)

HG = y + 8

Hence, We can formulate;

∠ E = ∠G

⇒ (7x - 15) + (7x - 15)  = 90°

⇒ 14x - 30 = 90

⇒ 14x = 60

⇒ x = 30/7

Hence., ∠ E = (7x - 5) = (7 x 30/7 - 5)

                                 = (30 - 5)

                                 = 25

Since, Opposite sides are equal in parallelogram.

Hence, We get;

3y = y + 8

2y = 8

y = 4

Hence, Measure of side FG is,

FG = 2 (y + 1)

FG = 2 (4 + 1)

FG = 10 units

Thus, Measure of angle E is, 25 degree

And, Measure of side FG is,

FG = 10 units

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The time series data should show a. trend b. cyclical c. seasonal, so the correct answer is d. all of the options are correct.

Time series data refers to a collection of data points that are measured over time at regular intervals. The time series data could refer to various metrics such as sales, revenue, foot traffic, website traffic, inventory levels, and other relevant business metrics that change over time.

By analyzing time series data, businesses can gain insights into trends, patterns, and other factors that can affect their operations.  

Time series can be used for forecasting. By analyzing past trends and patterns, businesses can make predictions about future performance and take proactive measures to achieve their goals.  

Since the time series will follow the trend, have multiple choices, and will be cyclical and it will analyze the patterns over time periods seasonally.

Therefore,

The time series data should show a. trend b. cyclical c. seasonal, so the correct answer is d. all of the options are correct.

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Answer:

A

Step-by-step explanation:

There is an x in all for terms that means that x = 0 is one of the factors and H is incorrect.

One of the remaining factors is x^2 - 9 which factors into (x - 3)(x + 3)

You are told that x = 1  is a zero which means x - 1 is a factor.

The answer is g(x) = x (x - 3)(x + 3) (x - 1) which is A

Answer:

The number of times the spinner will land on an odd number in 200 trials is 80 times

How to determine the number of times?

The sections on the spinner are given as

Sections = 1 to 7

From these sections, we have

Odd sections = 1, 3, 5 and 7

From the dot plot given, the total frequency of the odd sections is

Frequency of odd sections = 3 + 1 + 2 + 2

Evaluate

Frequency of odd sections = 8

Also, we have

Total frequency = 20

The number of times is then calculated as

Number of times = Frequency of odd sections/Total frequency * 200

So, we have

Number of times = 8/20 * 200

Evaluate

Number of times = 80

Hence, the number of times is 80 times

The temperature at the bottom of the mountain is 50.9 °C.

Let's work through the given information step by step to find the temperature at the bottom of the mountain.

The temperature in the hotel is 21 °C.

The temperature in the hotel is 26.7 °C warmer than at the top of the mountain.

Let's denote the temperature at the top of the mountain as T_top.

So, the temperature in the hotel can be expressed as T_top + 26.7 °C.

The temperature at the top of the mountain is 3.2 °C colder than at the bottom of the mountain.

Let's denote the temperature at the bottom of the mountain as T_bottom.

So, the temperature at the top of the mountain can be expressed as T_bottom - 3.2 °C.

Now, let's combine the information we have:

T_top + 26.7 °C = T_bottom - 3.2 °C

To find the temperature at the bottom of the mountain (T_bottom), we need to isolate it on one side of the equation. Let's do the calculations:

T_bottom = T_top + 26.7 °C + 3.2 °C

T_bottom = T_top + 29.9 °C

Since we know that the temperature in the hotel is 21 °C, we can substitute T_top with 21 °C:

T_bottom = 21 °C + 29.9 °C

T_bottom = 50.9 °C

Therefore, the temperature at the bottom of the mountain is 50.9 °C.

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Answer:

should be complementary angles

Answer:

m∠A ≈ 39.865°

Step-by-step explanation:

C² = 16² + 15² - 2 × 16 × 15 × cos (97°)

=> C = 23.22708085

sin (97°)/23.22708085 = sin A/15

=> sin A = sin (97°) × 15/23.22708085

=> A = \(sin^{-1}\)(sin (97°) × 15/23.22708085)

=> A ≈ 39.865

Answer: 48

Step-by-step explanation:

The Area of rectangle is 48 unit².

The total space occupied by a flat (2-D) surface or the shape of an object is defined as its area. The area of a plane figure is the space enclosed by its boundary. The number of unit squares that cover the surface of a closed form is the figure's area.

We have the graph from which we can write

width of rectangle = 8 unit

Length of rectangle= 6 unit

So, the area of rectangle

= lw

= 8 x 6

= 48 unit²

Thus, the Required Area is 48 unit².

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Answer:

3x^2 - x - 10

Step-by-step explanation:

Foil (First, outside, inside, last)

3x^2 + 5x + -6x - 10

combine like terms

3x^2 - x - 10

The largest ratio, among the following ratios: 5/36, 2:9, 3 to 18, 1:3 is 1:3.

The ratio refers to the relative size of one quantity compared to another.

The ratio, which is the quotient of two quantities or values, can be expressed as a decimal, percentage, or fraction.  We can also express the ratio in its standard form (:).

Given   Sum of     Equivalent

Ratio     Ratios          Ratios

5/36        36         13.89% or 0.1389

2:9           11          18.18% or 0.1818

3 to 18     21          14.28% or 0.14.28

1:3            4           25% or 0.25

Thus, we can conclude that 1:3 is the largest ratio amont the others.

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The side length of the cube is 5.6 feet (rounded to the nearest tenth).

We can calculate the side length of a cube with a volume of 141 cubic feet using the formula for cube volume , which is \(V = s^3\), where V is the volume and s is the side length.

We can calculate s by taking the cube root of both sides of the equation:

\(s = (V)^{(1/3)\)

Substituting V = 141, we get:

\(s = (141)^{(1/3)\)

By using a calculator to evaluate this expression, we may determine:

s ≈ 5.6

As a result, the cube's side length is roughly 5.6 feet (rounded to the closest tenth). This indicates that if we increase the side length by three, it will become longer. (\(s^3\)), we will get the volume of the cube, which is 141 cubic feet.

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9514 1404 393

Answer:

  45. (8√2)∠π/4

  46. (7√2)∠3π/4

Step-by-step explanation:

The polar form of a+bi is ...

  a +bi ⇔ (√(a²+b²))∠arctan(b/a)

__

45. 8 +8i = (√(8² +8²))∠arctan(8/8) = (8√2)∠π/4

__

46. -7 +7i = (√((-7)² +7²))∠arctan(7/-7) = (7√2)∠3π/4

_____

Additional comment

The arctan function usually returns a value in the range -π/2 to π/2. The quadrant of the desired result can be determined by the signs of the components of the vector. For negative real parts, π needs to be added to the value that is usually returned by the arctan function.

To find the middle term of the expansion of (2x/y^3)^12, we need to first determine the number of terms in the expansion.

Using the formula for the binomial theorem, we know that the number of terms in the expansion of (2x/y^3)^12 is 13.

Now, to find the middle term, we need to find the term that is in the middle of the expansion. Since there are 13 terms in the expansion, the middle term is the 7th term.

The general term of the expansion is given by:

T(r+1) = (12 C r) (2x)^r (y^-3)^(12-r)

So, the 7th term is:

T(8) = (12 C 7) (2x)^7 (y^-3)^(5)

Simplifying this expression, we get:

T(8) = (792) (128x^7 / y^15)

Therefore, the middle term of the expansion is 792(128x^7/y^15)

It is always a good practice to simplify the result whenever possible, but in this case, V14 is the simplest form of the product of V7 and V2.

To multiply the square roots V7 and V2, we can combine the numbers inside the square roots and simplify the result.

V7 * V2 = V(7 * 2) = V14

Multiplying the numbers under the square roots, we get 7 * 2 = 14. Therefore, the product of V7 and V2 is V14.

This means that the square root of 14 is the result of multiplying V7 and V2. However, it is important to note that V14 cannot be further simplified because 14 does not have any perfect square factors.

In summary, the product of V7 and V2 is V14. It is worth mentioning that when multiplying square roots, we can multiply the numbers inside the square roots and keep the square root symbol intact, unless the numbers inside have perfect square factors that can be simplified further.

It is always a good practice to simplify the result whenever possible, but in this case, V14 is the simplest form of the product of V7 and V2.

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